An Illustrated Introduction to the Truncated Fourier Transform

TL;DR

This paper introduces the univariate Truncated Fourier Transform, a variation of the FFT that handles input vectors of arbitrary length, with detailed illustrations to enhance understanding.

Contribution

It provides an accessible, illustrated presentation of the univariate TFT, originally developed by Joris van der Hoeven, expanding its accessibility.

Findings

The TFT handles non-power-of-two input lengths.

Detailed illustrations improve understanding of the TFT.

The method extends FFT applicability to arbitrary input sizes.

Abstract

The Truncated Fourier Transform (TFT) is a variation of the Discrete Fourier Transform (DFT/FFT) that allows for input vectors that do NOT have length $2^n$ for $n$ a positive integer. We present the univariate version of the TFT, originally due to Joris van der Hoeven, heavily illustrating the presentation in order to make these methods accessible to a broader audience.